Huygens' Wavelets

Huygens' principle is usually stated in the following
way: each point on the wavefront of a sound wave gives rise to a Huygens' wavelet.
Each wavelet propagates outward as a spherical sound wave, and the superposition
of all of the wavelets creates the total sound wave. This is a very useful concept
for understanding diffraction and some other aspects of wave propagation. The
principle applies to electromagnetic and other waves as well, but the form of
the wavelet varies.

Equation G14 in the section on the General
Solution of the wave equation is the mathematical expression of this principle.
Actually equation G14 shows that a more general statement of the principle is
that each point on any surface gives rise to a wavelet - not just an
equiphase surface. However the equations are simpler for an equiphase surface,
and this is the conventional definition, so that will be assumed here.

Mathematical Form of the Wavelet

We consider the specific case of a plane wave propagating in the +z direction,
and the wavefront at the plane z=0. The surface of integration for equation
G14 is the plane at z=0 and a hemisphere of infinite radius on the +z side of
the plane. The velocity u and pressure p (at a given instant
in time), and the normal n at the wavefront are specified to
be

The vector Rp extends from the origin to the evaluation
point, and θ is the angle between Rp
and the z-axis. The vector R extends from the origin to the
source. (see the geometry figure). Here we define the
wavelet source location to be at the origin, so R=0, and r=-Rp.
The normal n points outside the volume enclosed by the surface
of integration, not in the direction of wave propagation. The form of the wavelet
can be obtained directly from equation G14 for the pressure term, and by taking
the gradient for the velocity term. The results are

There are several interesting things about these
equations. First, the equation for velocity u(Rp)
is actually a bit more complicated than the Huygens' wavelet for an electromagnetic
wave. This is surprising since sound is generally viewed as a simpler process.
Second, the waves only become truly spherical in the far-field, and are pretty
complicated in the near field. In the far-field the overall shape of the Huygens'
wavelet is proportional to 1+cosθ, for both pressure and velocity, exactly
the same as the electromagnetic case. This is only true if the ratio p0/Z0=u0.
At a boundary between media of different acoustic impedances this is not true,
and a reflected wave is created as well as a forward propagating wave.

To me the most startling aspect of these equations
is that the velocity component u0 of the source at the origin radiates
isotropically, whereas the pressure component p0 has a directional
pattern! Physically, higher pressure means a higher molecular density and higher
temperature, which is totally lacking in directionality. Conversely the velocity
is directional. (This odd directivity could be simply due to the fact
that a surface and a normal are specified as well as pressure and velocity.
I intend to look into this in the future. Which I did; see two paragraphs below).

This seemed so counterintuitive that I felt compelled
to double check the equations. For the incoming plane wave I considered a spherical
surface centered at the origin. I used the values of velocity and pressure on
the surface in equation G14, and in the equation derived from the gradient of
G12, and numerically computed the sound pressure and velocity inside, and outside,
the surface. Inside the surface I got the plane wave, and outside the surface
I got zero, exactly as expected. Note that the wavelets in this case are more
complicated than the above equations, since the sphere is not an equiphase surface,
but the above equations are a special case of the more general wavelets. This
numerical test case is also counterintuitive; the superposition of the wavelets
create a plane wave inside the sphere, but add up to zero everywhere outside
the sphere. Mathematically this is the correct behavior, but a first it was
difficult for me to imagine real physical sources resulting in this situation.

Then I considered the following mind experiment: fill all space with small
tubes of square cross section and vanishingly thin walls. Excite waves in all
tubes such that all space has exactly the same wave values as would exist with
a plane wave in free space. Now cut a circular cavity out of the mass of tubes.
All boundary conditions are satisfied if the cavity contains a plane wave. Thus
the fields at the end of the cut tubes are acting exactly like the sources in
the computer model - radiating a plane wave inside the cavity and not radiating
at all outside.

With a little more thought the resolution of the
directivity "paradox" is also obvious. Consider a point source of
pressure. Enclose the source in a small sphere. Each point on the spherical
surface radiates a wavelet with a directional pattern, with a maximum in the
direction of the surface normal. But integrating over the entire sphere, the
normals point equally in all directions, so the sum of the wavelets produce
an isotropic pattern. Alternatively, for a point source of velocity directed
towards the north pole, each point on the sphere radiates a wavelet with an
isotropic pattern. But the north pole wavelet has a positive magnitude, the
south pole negative, and the magnitude is zero at the equator. The sum of the
wavelets produces a dipole pattern. So the directivity paradox exists only if
one wavelet is considered in isolation.

Power Flow

Next I computed
the power flow, which is shown in this strange and fascinating
figure [68 kb]. When I originally computed this figure and saw a jet of
power flowing out of the origin in the negative z-direction, I immediately concluded
that I must have made a sign error someplace. I have now exhaustively checked
the equations and I believe the figure is correct.

The solid black lines are parallel to the direction
of power flow. The blue line segments are vectors parallel to the power flow,
with power flowing from the end with the black dot towards the end with the
red dot. For small radii the near-field power flow is dominated by two terms
that are inversely proportional to the fourth power of the radius. Even ignoring
the singularity at the origin this leads to huge power flows. The length of
each vector is proportional to the power flux, except that all vectors have
been scaled by a factor of radius to the 4th power, to eliminate these huge
variations.

The pattern in 3-D is this figure revolved around
the z-axis. Close to the origin the power flows in the form of a toroidal vortex.
The power is spit out in a jet in the negative z-direction, and sheds off the
vortex to flow predominantly in the positive z-direction. For
an incident plane wave propagating in the negative z-direction the sign of u0
is reversed, and the wavelet pattern flips around the x-axis. The power flux
at a large radius is proportional to either (1+cosθ)2 or (1-cosθ)2
depending on the sign of p0u0. Far away from the origin
the flow becomes radially outward in all directions, in either case.

There is another pair of Huygens' wavelets generated
by the conjugate Green's function (the complex conjugate of
Equation G2). Far away from the origin the power flow becomes radially inward in all
directions. For an incident plane wave traveling in the positive z-direction
the pattern shown in the figure is flipped around the x-axis, and the direction
of power flow is reversed everywhere except inside the toroidal vortex. For
the top part of the vortex shown in the figure the flow is still clockwise.

So the Huygens' wavelet equations are apparently
mathematically valid. Do they have a physical significance? Obviously sound
cannot really emanate from a point, so that is one limit to any physical interpretation.
But for a one meter wavelength the size of the vortex is roughly 10 cm, so there
is no lack of molecules until you get very close to the origin. Huygens' wavelets
are a valid solution of the sound wave equations in all space for ρ
> 0, and mathematical sources exist for the wavelets. It is hard for me to
imagine molecules resulting in this peculiar behavior, but at this point I don't
know how to argue with the math.

Numerical Experiments

The behavior of Huygens' wavelets varies with the
nature of the incident field, and also depends on whether the regular or conjugate
Green's function is used to develop the general solution. To study this behavior,
a numerical model of the general solution was developed. The integration surfaces
are two concentric spheres, centered at the origin, with radii, set to 1.67
and 4.46 wavelengths for the results presented here. The incident field can
be a plane wave propagating in the plus or minus z-direction, or an incoming
or outgoing spherical wave. The origin of the spherical wave is on the z-axis,
and can be set outside the larger sphere or inside the smaller sphere. The inner
sphere is empty, except for the source, when it is present. The sound wave pressure
and velocity was computed in the volume V between the two spheres. The results
were probed along a line parallel to the z-axis, but with arbitrary x- and y-offsets.
In all cases the results agreed virtually perfectly with the incident field,
which is a very powerful confirmation of both the model and the general solution
equations that the model is based on.

The results are totally symmetrical for plane waves
propagating in +z and -z directions, and for spherical waves with origins on
the positive and negative parts of the z-axis. For simplicity, results are described
for plane waves propagating in the +z direction, and spherical waves with origins
on the negative part of the z-axis.

Only the surface of the larger sphere contributes to the results. Huygens'
wavelets radiate outward from the -z side of the sphere, and are absorbed by
the opposite side. Net result inside V is a wave traveling predominantly in
the +z direction.

2) Incoming spherical wave from a point outside larger sphere.

Only the surface of the larger sphere contributes to the results. Huygens'
wavelets radiate outward from the +z side of the sphere, and are absorbed by
the opposite side. Net result inside V is a wave traveling predominantly in
the -z direction.

3) Outgoing spherical wave from a point inside the smaller sphere.

Only the surface of the smaller sphere contributes to the results. Huygens'
wavelets radiate outward everywhere on the surface. Net result inside V is an
outward propagating spherical wave.

4) Incoming spherical wave from a point inside the smaller sphere.

This is where things get interesting. Both surfaces contribute to the results.
The Huygens wavelets from the outer surface combine to produce standing waves.
The Huygens' wavelets from the inner surface propagate outward. Net result inside
V is an inward propagating spherical wave.

Conjugate Green's function

Wavelets are incoming instead of outgoing, but the superposition of all of
the wavelets produce results that are identical to the regular Green's function
for situations 1) and 2).

3) Outgoing spherical wave from a point inside the smaller sphere.

Also interesting. Both surfaces contribute to the results. The Huygens' wavelets
from the outer surface combine to produce standing waves. The Huygens' wavelets
from the inner surface propagate inward. Net result inside V is an outward propagating
spherical wave.

4) Incoming spherical wave from a point inside the smaller sphere.

Only the surface of the smaller sphere contributes to the results. Surface
sucks in incoming wavelets.

Summary

In summary, although
equation G14 yields correct answers to real-world problems, at the level of
Huygens' wavelets I don't know how to connect the mathematics with real-world
molecules. Clearly the behavior of the wavelets is far more complex than
the simple picture of outward propagating spherical waves.
If anyone has additional thoughts on this, send
me an e-mail and I will append any comments I find interesting
- with full credit of course.